The story changes once we consider infinite locally finite graphs. Then the operator \(\Delta \)  may be represented as an infinite matrix. In order to study spectral properties, we need to restrict our attention to the Hilbert space \(l^2(V)\) , making \(\Delta \)  a bounded self adjoint operator. Then we can apply the classical theory of spectra of Hilbert space operators. Infinite locally finite graphs and their spectral properties have connections to group theory, the theory of random walks [1]}, visual processing (as was indicated by A.Agrachev) and mathematical physics [2]} [3]}.
